Terminology and notation

Let u\colon X\to Y be a continuous map between topological spaces. We say that u is proper if it is proper in the sense of Bourbaki (i.e. universally closed) and furthermore separated (i.e. the diagonal of X\times_Y X is closed).

Following Gabriel and Zisman, we say simplicial where we would have previously said semi-simplicial.

We denote by A a Noetherian subring of \mathbb{R} such that A\otimes\mathbb{Q} is a field. The useful cases are A=\mathbb{Z}, \mathbb{Q}, or \mathbb{R}.

A mixed Hodge A-structure consists of an A-module H_A of finite type, a finite increasing filtration W on the A\otimes\mathbb{Q}-vector space H_{A\otimes\mathbb{Q}}=H_A\otimes\mathbb{Q}, and a finite decreasing filtration F on the \mathbb{C}-vector space H_\mathbb{C}=H_A\otimes_A\mathbb{C}. We demand that the (\operatorname{Gr}_n^W(H_{A\otimes\mathbb{Q}}),\operatorname{Gr}_n^W(F)) be Hodge A\otimes\mathbb{Q}-structures. For A=\mathbb{Z} (resp. A=\mathbb{Q}), we recover (2.3.1) (resp. (2.3.8)); the results of §2.3 carry over as they are.

Let \mathscr{E} be a subset of \mathbb{Z}\times\mathbb{Z}. We say that a mixed Hodge structure H is of degree \mathscr{E} if the Hodge numbers h^{pq} are zero for (p,q)\not\in\mathscr{E}.

From now on, we denote by \Omega_X^\bullet(\log D) what we previously denoted by \Omega_X^\bullet\langle D\rangle (3.1.2).

Unless explicitly stated otherwise, scheme means “scheme of finite type over \mathbb{C}”, and a sheaf on a scheme X is a sheaf on the underlying topological space of X_\mathrm{an}.